85.12.6 problem 6

Internal problem ID [22500]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 40
Problem number : 6
Date solved : Thursday, October 02, 2025 at 08:43:07 PM
CAS classification : [[_homogeneous, `class D`]]

\begin{align*} y^{\prime }&=\frac {y+\cos \left (\frac {y}{x}\right )^{2}}{x} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=\frac {\pi }{4} \\ \end{align*}

✓ Maple. Time used: 0.103 (sec). Leaf size: 16

ode:=diff(y(x),x) = (y(x)+cos(y(x)/x)^2)/x; 
ic:=[y(1) = 1/4*Pi]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \arctan \left (\frac {2 x -1}{x}\right ) x \]

✓ Mathematica. Time used: 0.327 (sec). Leaf size: 15

ode=D[y[x],x]==  (y[x]+Cos[ y[x]/x ]^2)/x; 
ic={y[1]==Pi/4}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x \arctan \left (2-\frac {1}{x}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (y(x) + cos(y(x)/x)**2)/x,0) 
ics = {y(1): pi/4} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (y(x) + cos(y(x)/x)**2)/x cannot be solved by the lie group method